In the case of nonlinear systems, the identification problem is much more difficult. In the case that the structure of the system is known a priori, the identification problem can be reduced to a parameter estimation problem which can be solved using Lyapunov-based adaptive estimation methods [1]}. In the case where the structure of the system is unknown, some assumptions have to be made. For instance, in [2]} a set of basis functions are specified beforehand, such as constant, polynomial, and trigonometric functions, in the hope that the data can be decomposed appropriately with respect to the chosen basis. Another approach is using kernel dynamic mode decomposition (DMD) (c.f. [3]}) which aims to decompose a time series corresponding to a nonlinear dynamical system into a collection of dynamic modes using spectral decomposition of the Koopman operator. In order to establish convergence guarantees and error bounds, the Koopman operator needs to be compact [4]}, [5]}, which is typically not the case when the dynamics are nonlinear [6]}. While the techniques described above have been proven effective in many practical applications [2]},[3]}, [9]} , they generally do not provide theoretical error bound guarantees.
